Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s + 2·9-s − 6·11-s + 16-s + 2·19-s − 2·20-s − 25-s + 2·29-s − 12·31-s − 2·36-s + 5·41-s + 6·44-s + 4·45-s − 6·49-s − 12·55-s + 4·59-s − 8·61-s − 64-s − 2·76-s + 12·79-s + 2·80-s − 5·81-s + 24·89-s + 4·95-s − 12·99-s + 100-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 2/3·9-s − 1.80·11-s + 1/4·16-s + 0.458·19-s − 0.447·20-s − 1/5·25-s + 0.371·29-s − 2.15·31-s − 1/3·36-s + 0.780·41-s + 0.904·44-s + 0.596·45-s − 6/7·49-s − 1.61·55-s + 0.520·59-s − 1.02·61-s − 1/8·64-s − 0.229·76-s + 1.35·79-s + 0.223·80-s − 5/9·81-s + 2.54·89-s + 0.410·95-s − 1.20·99-s + 1/10·100-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4100\)    =    \(2^{2} \cdot 5^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4100,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.7902602434$
$L(\frac12)$  $\approx$  $0.7902602434$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;41\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;41\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good3$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$V_4$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$V_4$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$V_4$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$V_4$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$V_4$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$V_4$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$V_4$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$V_4$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
97$V_4$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.76805004435645432862239245801, −12.01358658782348319686332426916, −11.12838465475226465031138874472, −10.50181056919323422276502130066, −10.19524677105078893381228465769, −9.400484056125080834230342955772, −9.109403373605692684727508650678, −7.988358683254346292603082190295, −7.67909308622040811984694105441, −6.80904110843827372632476648277, −5.78628155331956279239987813546, −5.33426293676225140059122001280, −4.55099246561492137562690343487, −3.34495107937233360156966555355, −2.09231570919707459296302770195, 2.09231570919707459296302770195, 3.34495107937233360156966555355, 4.55099246561492137562690343487, 5.33426293676225140059122001280, 5.78628155331956279239987813546, 6.80904110843827372632476648277, 7.67909308622040811984694105441, 7.988358683254346292603082190295, 9.109403373605692684727508650678, 9.400484056125080834230342955772, 10.19524677105078893381228465769, 10.50181056919323422276502130066, 11.12838465475226465031138874472, 12.01358658782348319686332426916, 12.76805004435645432862239245801

Graph of the $Z$-function along the critical line